Eyeglass lens designing method and eyeglass lens

ABSTRACT

It is made possible to easily obtain a spectacle lens with higher performance in a spectacle lens designing method in which an eyeball motion (Listing&#39;s Law) is taken into consideration. A spectacle lens designing method in which an eye motion (Listing&#39;s Law) is taken into consideration, and which uses, as an evaluation function regarding visual acuity constituting a merit function which is used in optimization calculation, a visual acuity evaluation function (logMAR) derived in an ordinary manner from a visual acuity measured value V which is actually measured. Note that the visual acuity evaluation (logMAR) is represented by the following equation (1), letting a curvature of field be an ordinary aberration of a spectacle lens, and a residual astigmatism be an astigmatism extendedly defined from the spectacle lens designing in which the Listing&#39;s Law is taken into consideration. &lt;DF&gt;logMAR = log10(1/V(curvature of field, residual astigmatism)) &lt;/DF&gt; &lt;IMAGE&gt;

TECHNICAL FIELD

The present invention relates to a spectacle lens designing method and aspectacle lens designed by the same.

BACKGROUND ART

The Listing's Law in an eyeball motion means that, when an eyeball looksfar forward (first eye position), a rotation axis of the eyeball motionexists in a surface including the center of rotation of the eyeball andbeing perpendicular to this eye position (Listing's surface). In thiscase, when the eyeball rotates from the first eye position alongspectacle principal meridians (representing two vertical and horizontallines orthogonal to each other on a Gaussian curved surface andrepresenting the same below) according to the Listing's Law at the timeone wears astigmatic spectacles, the spectacle principal meridians andaxes of a coordinate system rotating according to the Listing's Lawbecome parallel to each other and an angle between them becomes 0.

However, when the eyeball motion changes in a direction different fromthe spectacle meridians, the angle made by the spectacle meridians andthe coordinate axes rotating according to the Listing's Law do notbecome 0 to cause an angle deviation.

By taking this angle deviation of the coordinate system intoconsideration, an accurate astigmatism and curvature of field (alsocalled a power error) can be calculated.

A spectacle lens designing method in which this eyeball motion(Listing's Law) is taken into consideration is disclosed in JapanesePatent Laid-open No. Sho 57-10112 (hereinafter, referred to as Prior art1)(refer to FIG. 5 in Prior art 1).

Meanwhile, optimization of evaluation functions for several kinds ofaberrations, a lens shape, and so on by optimization calculation in anaberration correction process in designing a lens is known as isdisclosed, for example, in Japanese Patent Publication No. Hei 2-38930.

To explain the outline of this optimization calculation, takingdesigning of a single vision aspherical lens for example, though it is aknown technique in spectacle lens designing, data on a lens material andprescription specifications are given as basic design specifications,items such as a center thickness are further included as additionalspecifications in a case of a positive lens, and a combination ofrefractive surface shapes of a front surface and a rear surface whichsatisfies them and has as small an optical aberration as possible isobtained by calculation. The refractive surface is expressed as asurface which is mathematized by a function and the function consists ofa plurality of parameters defining a spectacle lens. The parametersinclude a refractive index of the material, a lens diameter, radii ofcurvature of the front surface and the rear surface, the centerthickness, an aspherical conic coefficient, a high degree asphericalcoefficient, and so on. They are classified into fixed factors andvariable factors according to the object of the lens designing, and thevariable factors are dealt as variable parameters.

Then, using a ray tracing method and a wave front tracing method, aplurality of evaluation points whose distances from an optical axis onthe refractive surface are different are set on the lens surface, theoptical aberration on each of the evaluation points is expressed as anevaluation function (merit function), and calculation to obtain theminimum evaluation function is done using an optimization calculationmethod such as a damped least square method. At this time, simulationsare repeated while operating the variable parameters of the refractivesurface, and when a target value is obtained, the final shape of therefractive surface is determined.

As the parameters constituting the evaluation function (merit function)in the optimization calculation, an astigmatism and a curvature of fieldare generally known, and in a case, for example, when the front surfaceand the rear surface are both spherically designed in a designing methodin a prior art, assuming that the aberrations showing, in a unit ofdiopter, two focal positions Ft, Fs obtained by the ray tracing methodbased on a focus D obtained by a paraxial ray tracing are t (tangentialerror) and s (sagittal error) as shown in FIG. 11, a lens in which theastigmatism=(t−s) is minimum is called a Tscheming Form and a lens inwhich the curvature of field=(t+s)/2 is minimum is called a PercivalForm. In Japanese Patent Publication No. Sho 42-9416, an evaluationfunction in which t and s are complicatedly combined and which isexpressed as a horizontal aberration is disclosed.

A distortion aberration is known to be also an important evaluationfunction in the aforesaid design optimization calculation, and designingin which it is taken into consideration is proposed, for example, inJapanese Patent Laid-open No. Sho 55-59425 (hereinafter, referred to asPrior art 2) and APPLIED OPTICS, Vol. 21, No. 162982-2991: written byMilton Katz (hereinafter, referred to as Prior art 3).

As one of free curved surfaces among lens refractive surface shapes, anatoric surface is known besides a spherical surface and an astigmaticsurface. The use of a spline function as an equation used to express theatoric surface is disclosed in Japanese Patent Laid-open No. Sho62-30216 (Prior art 4) and an equation using orthogonal functions of xyis disclosed in International Publication No. WO 93/07525 (hereinafter,referred to as Prior art 5) is disclosed.

In recent years, however, it has been found out that visual acuity isclosely related to processing in the brain and it has been known thatthe visual acuity is mainly constituted by an image on a retina andprocessing of the image in the retina and the brain.

Meanwhile, in the designing of spectacle lenses in the prior art, suchan idea has been dominant that performance of a spectacle lens isimproved as optical performance of the lens becomes higher.

For example, in the optimization calculation method described above, theevaluation function in the prior art is based on an evaluation only byoptical calculation, such as evaluation of the size of an image and t(tangential error) and s (sagittal error) of the aberration and so onwhich are calculated at a far point sphere (FPS) in FIG. 11 by the raytracing method, and furthermore, an image plane or a retina surface arealso dealt as a film surface of a camera without considering aphysiological function of an eye such as the eyeball motion.

Furthermore, since the distortion aberration is dealt as an opticalamount of a camera as described above also in the above-mentioned Priorart 3, the evaluation function used in it is different from anevaluation function based on a visual angle magnification M which isused in spectacles (for example, KOHGAKU (OPTICS), Vol. 19, No. 10“Futatabi Kakubairitsu nitsuite (On Angle Magnification Again)” KazuoMiyake), and furthermore, an astigmatic lens and the designing in whichthe eyeball motion is taken into consideration are not disclosed either.Furthermore, the above-mentioned Prior art 2 does not disclose anyconcrete technical content thereof and its actual state is not clear.

Meanwhile, in lens designing, the use of the spline function for theatoric surface having a higher degree of freedom of expression, which isdisclosed in the above-mentioned Prior art 4, enables the expression offree surface shapes, but it has a disadvantage that it basically lacksprecision in surface expression. Moreover, in the above-mentioned Priorart 5, the properties of the eyeball motion using the Listing's Law arenot utilized to result in an insufficient optical surface.

Prior art 1 discloses a designing method in which the eyeball motion istaken into consideration using the Listing's Law. However, here, theexplanation of the above-described technical idea is focused on, and inthe concrete lens designing, performance evaluation is made based onlyon an astigmatism derived from optical calculation, and an evaluationfunction in the optimization calculation is insufficient.

Moreover, no concrete disclosure on the expression of a lens surface isgiven.

Furthermore, designing in this Prior art 1 is essentially the same asthe one in the prior art based on the idea that performance of aspectacle lens is improved as optical performance becomes higher and itgives no consideration to the correlation with visual acuity.

Thus, it is clear that performance evaluation of a spectacle lens basedonly on indexes such as an optical amount on the retina and theaberrations is inaccurate as a simulation on a living human body sinceno consideration is given to the viewpoints of the processing in theretina and the brain and of the eyeball motion as described above.

An object of the present invention, which is made to solve theseproblems, is to provide a spectacle lens with high performance whichimproves visual acuity and to provide a designing method of the same.

DISCLOSURE OF THE INVENTION

In order to solve the above-described problems, a first invention is

a spectacle lens designing method in which an eyeball motion (Listing'sLaw) is taken into consideration, and which is characterized in that amerit function used in optimization calculation processing of lensdesigning includes a visual acuity evaluation function (logMAR) derivedfrom a visual acuity measured value V,

where the visual acuity evaluation function (logMAR) is expressed by thefollowing equation (1), letting a curvature of field be an aberration ofa spectacle lens and a residual astigmatism be an astigmatism definedfrom spectacle lens designing in which the Listing's Law is taken intoconsideration:

the visual acuity evaluation function (logMAR)=log₁₀ (1/V (curvature offield, residual astigmatism))  (1)

A second invention is a spectacle lens designing method which ischaracterized in that, in the spectacle lens designing method of thefirst invention, letting the visual acuity measured value V beV=2^(−x·K) (where K={(residual S diopter+residual C diopter/2)²+(residual C diopter/2)²}^(1/2) and X is a coefficient between 0.5 and 2according to actual measurement data), the visual acuity evaluationfunction (logMAR) is expressed by the following equation (2) which is anapproximate equation:

the visual acuity evaluation function (logMAR)=X×log₁₀2×{curvature offield²+(residual astigmatism/2)²}^(1/2)  (2)

A third invention is a spectacle lens designing method which ischaracterized in that, in the spectacle lens designing method of thefirst invention, the merit function includes an evaluation function on adistortion aberration (residual distortion aberration DIST) and theevaluation function is expressed by the following equation (3):

residual distortion aberration DIST=Sign×100×(absolute value of residualvisual angle magnification/absolute value of central visual anglemagnification M ₀)   (3)

where:

the residual visual angle magnification is the distortion aberrationdefined from the spectacle lens designing in which the Listing's Law istaken into consideration; and

Sign is a positive/negative sign.

A fourth invention is a spectacle lens designing method which ischaracterized in that, in the lens designing method according to any oneof the first invention to the third invention, the merit function isused in optimization calculation of lens designing of a bi-asphericallens in which a front surface is an axially symmetrical asphericalsurface and a rear surface is an aspherical surface expressed by thefollowing equation (4): $\begin{matrix}{{Z2} = {{c(\theta)} \cdot {r^{2}/( {1 + \sqrt{ {1 - {( {1 + {k(\theta)}} ) \cdot {c(\theta)}^{2} \cdot r^{2}}} )} + {\sum\limits_{n}{{a( {n,\theta} )} \cdot r^{n}}}} }}} & (4)\end{matrix}$

where:

c(θ), k(θ) are functions for an azimuth θ;

a(n, θ) is a function for an n degree of a distance r and the azimuth θ;

as for a definition domain of the azimuth θ, 0 degree to 90 degreesrepresents 0 degree to 360 degrees due to plane symmetry of anastigmatic lens;

c(θ) is a curvature of a lens center and is expressed by the followingequation (5) based on the Euler's theorem, letting a curvature of aspectacle principal meridian in the Gaussian curve theorem be c(0) at 0degree and c(90) at 90 degrees. In this case, 0 degree is a sphericaldiopter axis and 90 degrees is a cylinder diopter axis;

c(θ)=c(0)·cos² θ+c(90)·sin²θ  (5)

k(θ), which is similar to c(θ) above, represents an equation in whichthe sign c is replaced by the sign k in the above equation (5); and

a(n, θ) satisfies requirements of plane continuity and plane symmetry,is a surface further satisfying a requirement of a surface which iscapable of controlling an aberration due to an angle deviation whichoccurs due to the Listing's Law, and further satisfies the followingconditions {circle around (1)} to {circle around (4)}:

{circle around (1)}: having a functional relation of the azimuth θ from0 degree to 90 degrees;

{circle around (2)}: a linear differential coefficient of the azimuth θis 0 from 0 degree to 90 degrees;

{circle around (3)}: a high degree differential coefficient iscontinuous; and

{circle around (4)}: having a control parameter group Ps(n) which iscapable of controlling a value of a(n, θ) at an angle θ of a functionbetween the azimuths 0 degree and 90 degrees (where 1 to 3 arepreferable for s, and n signifies a degree in the above equation (4)).

A fifth invention is a spectacle lens designing method which ischaracterized in that, in the spectacle lens designing method accordingto the fourth invention, a(n, θ) in the above equation (4) is expressedby the following equation (6) which is a quartic polynominal of theazimuth θ, letting a be a(n, 0), a(n, 45), and a(n, 90) when the azimuthθ is 0 degree, 45 degrees, and 90 degrees respectively:

a(n, θ)=a(n, 0)+(−11

·a(n, 0)+16

·a(n, 45)−5

·a(n, 90))

·θ²/(4·90²)+(9

·a(n, 0)−16

·a(n, 45)+7·a(n, 90))

·θ³/(4·90³)+(−2

·a(n, 0)+4·a(n, 45)−2

·a(n, 90)·θ⁴/(4

·90⁴)  (6)

where a control parameter is one for the degree n of the distance r fromthe center and a control parameter P1(n) is a(n, 45).

A sixth invention is a spectacle lens designing method which ischaracterized in that, in the spectacle lens designing method accordingto the fourth invention, a(n, θ) in the above equation (4) is expressedby the following equation (7), letting a be a(n, 0) and a(n, 90) whenthe azimuth θ is 0 degree and 90 degrees respectively:

a(n, θ)=a(n, 0)·cos² θ+a(n, 90)·sin² θ+P 1(n)·sin²(2·θ)  (7)

where a control parameter is one control parameter P1(n) for the degreen of the distance r from the center.

A seventh invention is a spectacle lens which is characterized in thatit is designed by the spectacle lens designing method according to anyone of the first invention to the sixth invention.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is an explanatory view of a spectacle lens designing methodaccording to an embodiment of the present invention;

FIG. 2 is an explanatory view of an extended DIST;

FIG. 3 is a view showing Table 1 in which lens data in Example 1 arelisted;

FIG. 4 is a view showing Table 2 in which lens data in Comparisonexample of Example 1 are listed;

FIG. 5 is a view showing the visual acuity evaluation function (logMAR)in Example 1;

FIG. 6 is a view showing the visual acuity evaluation function (logMAR)in Comparison example of Example 1;

FIG. 7 is a view showing Table 3 in which lens data in Example 2 arelisted;

FIG. 8 is a view showing the distribution of a first quadrant of theextended DIST in Example 2;

FIG. 9 is a view showing the distribution of the extended DIST inComparison example of Example 1;

FIG. 10 is a view showing the visual acuity evaluation function (logMAR)in Example 2;

FIG. 11 is an explanatory view of a spectacle lens designing method in aprior art; and

FIG. 12 is a view showing actual measurement values of visual acuity.

VS . . . rear vertex spherical surface; V . . . rear vertex; W . . .reference point of focal length; R . . . center of rotation of eyeball;FPS . . . far point sphere; Ft . . . focus in radial tangent direction;Fs . . . focus in sagittal direction; D . . . image on far point sphere;Ws . . . reference point of focal length of ray passing on S axis; Wc .. . reference point of focal length of ray passing on C axis; FPS: farpoint sphere in S axis direction; FPC: far point sphere in C axisdirection; Fst: focus in S axis direction of ray passing on S axis; Fss. . . focus in C axis direction of ray passing on S axis; Fct . . .focus in S axis direction of ray passing on C axis; Fcs . . . focus in Caxis direction of ray passing on C axis; DS . . . image on far pointsphere in S axis direction; DC . . . image on far point sphere in C axisdirection; P . . . visual angle magnification evaluation point; MO . . .reference visual angle magnification in P direction; M . . . visualangle magnification at position P

BEST MODE FOR CARRYING OUT THE INVENTION

As a paper on retina and brain processing regarding visual acuity,Optmetric Monthly, November: 31-32 1981: written by Robert N. Kleinstein(hereinafter, referred to as Paper 1) is available.

A drawing in the above Paper 1 shows a view in which a visual acuitymeasured value is expressed by a fraction visual acuity value, taking Sdiopter and C diopter as spectacle terms in a horizontal axis and avertical axis respectively and an experiment of measuring visual acuityof a spectacle wearer with his/her spectacles taken off is conducted. Inorder to use this Paper 1 as an evaluation function of a merit functionin spectacle lens designing, the measured values are modified in such amanner that the signs of the horizontal axis value S and the verticalaxis value C are reversed, namely, the residual S diopter and theresidual C diopter are taken in the horizontal axis and the verticalaxis respectively to obtain evaluation data showing how the visualacuity decreases when a subject person having normal visual acuity wearsspectacles with an aberration, reversely to the above experiment.

In FIG. 12 described above, data for the age of 5 to 15, 25 to 35, and45 to 55 are provided as actual measurement data, but since it ispreferable to use a virtual visual acuity measured value not affected byan adjusting power, the data for the age of 45 to 55 were used fromPaper 1 for convenience sake.

The residual S diopter and the residual C diopter mentioned above arecorrelated to an astigmatism and a curvature of field derived fromoptical calculation as described later. In the spectacle lens designingin the prior art in which the Listing's Law is not taken intoconsideration, however, the astigmatism and the curvature of fieldcannot be calculated accurately in regions in which an eyeball does notrotate along two spectacle principal meridians as previously described.Therefore, a spectacle lens designing system in which the Listing's Lawis taken into consideration and which includes new lens aberration(astigmatism and curvature of field) calculation is required in order touse the measured values of the visual acuity measurement in Paper 1mentioned above as an evaluation function on the entire surface of alens.

(Spectacle Lens Designing System Including Lens Aberration (Astigmatismand Curvature of Field) Calculation)

FIG. 1 is a view explaining one model to be a factor in a spectacle lensdesigning method according to an embodiment of the present invention,and FIG. 11 is a view explaining a model in a prior art with which theabove model is compared.

In a case of rays passing S and C axes of an astigmatic lens shown inFIG. 1, calculation similar to the case shown in FIG. 11 of a designingsystem in the prior art is valid.

However, on an axis in a lens radiation direction other than the S and Caxes of the astigmatic lens in FIG. 1, it is necessary to calculate theastigmatism and the curvature of field with an eyeball motion taken intoconsideration, which are calculated by the following method.

Hereinafter, the correlation of the residual S diopter and the residualC diopter with the astigmatism and the curvature of field in thespectacle lens designing system in which the Listing's Law is taken intoconsideration will be simply explained.

I. (Astigmatism and Curvature of Field)

In Prior art 1 in which the Listing's Law is taken into consideration,when the rotation is in a different direction from the spectacleprincipal meridians, the angle between the spectacle principal meridiansand coordinate axes rotating according to the Listing's Law does notbecome 0. When the angle deviation as described in the above Prior art 1occurs, the astigmatism, even when, typically, it is an astigmatismhaving an absolute value of the astigmatism equal to an absolute valueof a reference astigmatism (an astigmatic amount and a cylinder axis atthe center of a lens), has a direction like a vector value so that aresidual astigmatism whose value is not 0 newly occurs.

As for a calculation method of the above residual astigmatism, methodsof calculating an astigmatic lens and of the residual astigmatism of theastigmatic lens as disclosed in, for example, Prior art 1 areapplicable.

Meanwhile, the curvature of field as another factor does not change dueto the coordinate change according to the Listing's Law since thecurvature of field is a scalar amount not related to a vector.

I-1. (Residual Astigmatism)

Therefore, the correlation of the aforesaid residual astigmatism andcurvature of field with the residual S diopter and the residual Cdiopter is as follows:

(1) When the residual astigmatism is positive, their correlation isexpressed by the following equations (a), (b):

residual S diopter=curvature of field−residual astigmatism/2  (a)

residual C diopter=residual astigmatism  (b)

(2) When the residual astigmatism becomes negative in opticalcalculation, their correlation is expressed by the following equations(c), (d) based on an idea similar to diopter conversion of spectaclessince the residual C diopter is defined as positive:

residual S diopter=curvature of field+residual astigmatism/2  (c)

residual C diopter=−residual astigmatism  (d)

II. (Deriving Merit Function in Which Nonlinear Nature of Living HumanBody in View of Optical Performance is Taken into Consideration)

On analyzing FIG. 12 in the aforesaid Paper 1, it is first found outthat the horizontal axis (residual S diopter) is not symmetrical withrespect to the origin. Furthermore, the vertical axis (residual Cdiopter) has also nonlinear data peculiar to the living human body.

For example, when visual acuity values with the same absolute value onthe horizontal axis and with different signs are examined, it is clearthat the functional relation is not simple. Therefore, when optimizationcalculation is directly done in the optical calculation without takingthe nonlinear nature peculiar to the living human body intoconsideration, this does not always indicate that visual acuity througha designed lens is improved since the visual acuity value is nonlinearrelative to an optical performance value.

Therefore, in the embodiment of the present invention, an interpolationfunction V of fraction visual acuity is first prepared from the data onthe fraction visual acuity measured values in FIG. 12. Concretely, anequation (e) by which the interpolation function V can be calculatedeven with continuous residual S diopter and residual C diopter isprepared using a generally known interpolation method, taking the visualacuity values for horizontal axis values (residual S diopter) andvertical axis values (residual C diopter) by discrete values (every 0.1to 1 diopter).

This is expressed by the following equation:

interpolation function V=V(residual S diopter, residual C diopter)  (e)

Using this interpolation function V, the aforesaid residual astigmatismand curvature of field of the lens are calculated, and they aresubstituted for the residual S diopter and the residual C diopter in theequations (a), (b) or the equations (c), (d).

Then, the optical value and the visual acuity value are correlated insuch a manner that a right side is obtained by the optical calculationand a left side is the visual acuity value by actual measurement as inthe following equation (f):

interpolation function V=V(curvature of field, residualastigmatism)  (f)

The equation (f) in this state can be used as an evaluation function,but since nonlinearity is high, it is not the best state for theoptimization calculation.

Therefore, it is further transformed to the following equation (g)expressed by a visual acuity evaluation function logMAR, which is adefinition equation for representing visual acuity.

the visual acuity evaluation function (logMAR)=log₁₀(1/V(curvature offield, residual astigmatism))  (g)

Through the above processes, the evaluation function in which thenonlinear nature of the living human body from the optical performancepoint of view is taken into consideration is derived.

The visual acuity of the living human body of course changes to a greatextent depending on age, a measurement environment, and so on.

In actual application, however, the above-described basic methodrequires a large calculation amount in the optimization calculation.

Therefore, instead of the equation (e) by which the aforesaidinterpolation function V can be calculated, simple approximate equationssuch as the following equations (h), (i) can be used:

V=2^(−X·K)  (h)

where, ·K is expressed by the following equation (i):

K={(residual S diopter+residual C diopter/2)²+(residual Cdiopter/2)^(1/2)}  (i)

·X is a coefficient between 0.5 and 2 according to actually measureddata.

In the above case, V may be used as the evaluation function as it is,but the correlation with the visual acuity evaluation function logMAR isexpressed by the following equation, as explained in the aforesaid basicmethod.

the visual acuity evaluation function (logMAR)=X×log₁₀2×{(curvature offield²+(residual astigmatism/2)²)^(1/2)  (j)

Furthermore, the approximate equations can be transformed by includingmeasured values according to age besides data in the material foractually measured visual acuity and by using other visual acuitymeasurement data. For example, the transformation of the equation (h)such as the following equation V=3^(−K) is possible under the conditionof within a variable range of X. In this case, the equation (j) becomesas follows:

the visual acuity evaluation function (logMAR)=log₁₀3×{(curvature offield²+(residual astigmatism/2)²)^(1/2)

III. (Distortion Aberration with the Listing's Law Taken intoConsideration)

Furthermore, as an aberration to be corrected for spectacles, which isnot related to a visual acuity value, there is a distortion aberration.

This is widely known as a cause of sway and distortion occurring mainlyat the beginning when one starts to wear spectacles. Conventionally, thedistortion of spectacles is expressed as a visual angle magnification(for example, refer to KOHGAKU (OPTICS), Vol. 19, No. 10“FutatabiKakubairitsu nitsuite (On Angle Magnification Again)” written by KazuoMiyake, and so on).

When this is expressed by an equation, letting a central visual anglemagnification be M₀, the following equation (k) is obtained:

central visual angle magnification M ₀=1im _(exit angle→0)(tan(exitangle)/tan(incident angle))  (k)

Here, the central visual angle magnification M₀ can be easily calculatedby paraxial optical calculation. The central visual angle magnificationM₀ will be simply explained. When an emergent ray passes the center ofeyeball entrance pupil, the central visual angle magnification M₀ iscalled a spectacle magnification.

Further, letting a peripheral visual angle magnification be M, thisvisual angle magnification M is expressed by the following equation (l):

peripheral visual angle magnification M=tan(exit angle)/tan(incidentangle)  (1)

Then, the distortion aberration (DIST) of the spectacles is expressed bythe following equation (m) based on the equations (k), (l):

distortion aberration DIST=100×((M/M ₀)−1)  (m)

Incidentally, in the model in FIG. 1, the emergent ray passes the centerof rotation of the eyeball and the distortion aberration DIST is calleda dynamic distortion aberration of the spectacles.

Here, on studying the equation (m), a residual distortion aberrationDIST occurs due to the difference (angle deviation) of an axis directionsince the distortion aberration DIST, even when it is the aberrationDIST with the same amount, is a vector value, similarly to the previousexplanation on the astigmatism.

Therefore, the central visual angle magnification M₀ and the peripheralvisual angle magnification M in the prior art are calculated as thedistortion aberration DIST when they are in the same direction.

For example, if the central visual angle magnification M₀ and theperipheral visual angle magnification M in the same direction are thesame amount, the distortion aberration DIST is calculated as thedistortion aberration DIST=0 by the equation (m).

Since the aforesaid angle deviation caused by the eyeball motion isincluded in the calculation, the central visual angle magnification M₀and the peripheral visual angle magnification M are both extendedlydefined as vector amounts.

Then, when the lens is an astigmatic lens, the rotational visual anglemagnification M₀ becomes a vector value having a different value in theradiation direction at a lens diopter reference point (usually, thecenter part of the lens).

When a residual visual angle magnification is defined as a valueobtained by subtracting the central visual angle magnification from theperipheral visual angle magnification M, this residual visual anglemagnification is expressed by the following equation:

residual visual angle magnification=peripheral visual anglemagnification M−central visual angle magnification M ₀

The extended definition of the distortion aberration of the spectaclesaccording to the embodiment of the present invention in which theListing's Law is taken into consideration becomes the followingequations (n), (o):

residual visual angle magnification=peripheral visual anglemagnification M−central visual angle magnification M ₀  (n)

residual distortion aberration DIST=Sign×100×(absolute value of residualvisual angle magnification/absolute value of central visual anglemagnification M ₀)   (o)

where Sign is defined as a positive/negative sign of a scalar product ofthe residual visual angle magnification and the central visual anglemagnification M₀.

FIG. 2 is a view showing the correlation of the equations (n) and (o).

Through the above, a residual distortion aberration equation of thespectacles in which the Listing's Law is taken into consideration isderived and it is further incorporated in the merit function.

IV. (Preparation of Merit Function)

In the spectacle lens designing method according to the embodiment ofthe present invention, the state in which a ray passes a lens is assumedand simulation calculation is done by the ray tracing method, andusually, about 5 to about 10 axially symmetrical lenses can be adoptedand about 15 to about 10000 lenses according to this embodiment can beadopted to calculate the aforesaid equations (g), (o).

In the case of the aforesaid equation (g), different values are obtaineddepending on the evaluated object distance. Determination on whichobject distance is to be taken is made in consideration of a lenscharacteristic and so on.

For example, strictly speaking, there is no actually measured visualacuity value of near vision in an equation (p) described later, butresponses to the residual S diopter and the residual C diopter can becalculated assuming that they are similar to those in a case of farvision.

Furthermore, it is said that the dynamic distortion aberration of thespectacles is not related to the object distance theoretically, butactually, no clear material exists on how to deal with the distributionof the visual acuity and the distortion, and so on. Therefore, they canbe freely set within a range not departing from the object of thedesigning.

From the above, the merit function according to the present invention,which is a combined function of evaluation functions and is a singleevaluation criterion, becomes the following equation (p).$\begin{matrix}{{{{merit}\quad {function}} = {{a \times {\sum\limits_{n}( {{u_{n} \cdot {far}}\quad {vision}\quad \log \quad {MAR}_{n}} )^{2}}} + {b \times {\sum\limits_{n}( {{v_{n} \cdot {near}}\quad {vision}\quad \log \quad {MAR}_{n}} )^{2}}} + {c \times {\sum\limits_{n}( {{w_{n} \cdot {residual}}\quad {DIST}_{n}} )}}}}\quad} & (p)\end{matrix}$

Here, a, b, c are weight distribution of respective evaluationfunctions; u, v, w are weight distribution at respective evaluationpoints; and n is a lens evaluation point. Of course, the idea (=notadopted) that the weight distribution is 0 (zero) is included, butnaturally, they never become 0 synchronously.

However, few objective experimental data which determines the weight isavailable, and in actual application, the weight distribution is carriedout in consideration of the object of using the lens, and aesthetic,economical, optical factors and so on.

Moreover, it is possible to add to the merit function of the presentinvention items not directly related to the visual acuity such as a lensform and so on.

The aforesaid merit function (p) is made optimized using theoptimization method. This optimization method is as explained in thesection of the background art previously described (for example, theaforesaid Japanese Patent Publication No. Hei 2-38930 and so on).

The aforesaid merit function (p) will be studied from the viewpoint ofthe degree of freedom of designing a lens refractive surface.

When a front surface and a rear surface of the lens are free curvedsurfaces which can be transformed freely under the restrictive conditionthat the diopter of the lens is fixed based on a prescription value, afirst term or a second term in the merit function can be made zero bythe transformation of these two surfaces.

Specifically, at a certain object distance, the astigmatism and thecurvature of field which are constituent factors of the visual acuityevaluation function logMAR can both be made 0.

However, when an aesthetic factor of its appearance is added and aneconomical viewpoint such as manufacturing cost is taken intoconsideration in designing the front surface which is a surface on anobject side of the lens, for example, when the restrictive condition ofan axially symmetrical aspherical surface is added, it is difficult tosynchronously make the residual astigmatism and the curvature of field 0on the entire surface of the spectacle lens at a certain objectdistance.

Still more, it is generally difficult to make the residual distortionaberration DIST 0 in the surface structure where lens diopter exists,without influencing other evaluation functions. Therefore, a coefficientand weighting are dealt as design items. Furthermore, from the viewpointof the degree of freedom of designing, when the structure of the frontsurface is fixed, for example, by the condition of a sphere and so on,the degree of freedom of designing is restricted, and it becomesdifficult to control a third term in the merit function, namely, theresidual distortion aberration DIST.

In other words, the merit function is a function in which theaberrations are complicatedly combined as describe above, and if thesurface has a restriction such as a sphere when the merit function isoptimized by the optimization, the optimization is influenced by therestriction.

Therefore, it is preferable that the front surface and the rear surfaceof the spectacle lens are both set in such a manner that they can bedesigned by free transformation, thereby enabling the merit function tobe freely controlled and increasing the degree of freedom of designing.

V. (Design of Bi-aspherical Lens)

Here, as a design example in which the degree of freedom of designing istaken into consideration, the explanation will be given on a spectaclelens consisting of aspherical surfaces on both sides, which enables theabove merit function to be optimized by the optimization calculationwith high precision and with high calculation efficiency.

Since according to the Listing's Law, the rotation is made in aradiation direction from the first eye position of the eye as ispreviously described, a corresponding expression of a lens surfacebecomes directly corresponding to the eyeball motion when it isexpressed by a spherical coordinate system and a cylindrical coordinatesystem with the lens center being the origin.

However, when it is expressed by other coordinate systems, for example,an orthogonal coordinate system and so on, a high degree coefficientbecomes necessary, though they are mathematically equivalent, in orderto bring about an equivalent effect in numerical calculation, andconsequently, a calculation error is increased.

Furthermore, though the aforesaid spline curved surface, a NURBS curvedsurface, and so on are also capable of expressing very free curvedsurfaces, they are basically the orthogonal coordinate system similarlyto the above so that the similar problem occurs in the numericalcalculation.

Therefore, in this embodiment, an aspherical surface equation of thecylindrical coordinate system is used as a preferable method (refer to,for example, Prior art 2 for the aspherical surface equation of thecylindrical coordinate system in detail).

(Aspherical Surface Equation Expressing Refractive Surface Shape ofFront Surface)

A lens height Z1 of the front surface, which is expressed by thefollowing equation (q), is expressed as an equation of a lens crosssection. $\begin{matrix}{{Z1} = {c \cdot {r^{2}/( {1 + \sqrt{1 - {( {1 + k} ) \cdot c^{2} \cdot r^{2}}} + {\sum\limits_{n}{{a(n)} \cdot r^{n}}}} }}} & (q)\end{matrix}$

In the first term of the right side, which is a rotational quadricsurface; c is a center curvature; k is a conic coefficient; and r is adistance between the position of the lens projected on a horizontalplane of the cylindrical coordinate system and the origin, and in thesecond term, which is a deviation from the rotational quadric surface,n, though it takes values from 2, usually takes values from 4 to 12since it interferes with the first term. a(n) is an n degree coefficientof r and is an amount called an aspherical coefficient.

V-1 (Aspherical Surface Equation Expressing Refractive Surface Shape ofRear Surface)

An equation of the rear surface of the present invention is thefollowing equation (r): $\begin{matrix}{{Z2} = {{c(\theta)} \cdot {r^{2}/( {1 + \sqrt{ {1 - {( {1 + {k(\theta)}} ) \cdot {c(\theta)}^{2} \cdot r^{2}}} )} + {\sum\limits_{n}{{a( {n,\theta} )} \cdot r^{n}}}} }}} & (r)\end{matrix}$

Here, c(θ), k(θ) are functions for an azimuth θ. a(n, θ) is a functionfor the n degree of the distance r and the azimuth θ. Due to planesymmetry of the astigmatic lens, as for a definition domain of theazimuth θ, 0 degree to 90 degrees can represent 0 degree to 360 degrees.Here, c(θ) is a curvature of the lens center, and letting the curvatureof the two principal meridians orthogonal to each other be c(0) andc(90) at 0 degree and 90 degrees respectively, as is stated in theGaussian curve theorem, the following equation (s) is obtained from theEuler's theorem.

In the case of the lens, 0 degree and 90 degrees are taken in thespherical diopter axis and in the astigmatic diopter axis respectively,and c(θ) is expressed by the following equation (s):

c(θ)=c(0)·cos² θ+c(90)·sin² θ  (s)

k(θ) is similar to the above equation (s) and becomes an equation inwhich the sign c in c(θ) is replaced by the sign k.

a(n, θ) satisfies requirements of plane continuity and plane symmetry,is a surface further satisfying a requirement of a surface which iscapable of controlling an aberration due to an angle deviation whichoccurs due to the Listing's Law, and satisfies the following conditions{circle around (1)} to {circle around (4)}:

{circle around (1)}: having a functional relation of the azimuth θ from0 degree to 90 degrees;

{circle around (2)}: a linear differential coefficient of the azimuth θis 0 from 0 degree to 90 degrees;

{circle around (3)}: a high degree differential coefficient iscontinuous; and

{circle around (4)}: having a parameter group: Ps(n) which is capable ofcontrolling a value a(n, θ) at an angle θ of a function between theazimuths 0 degree and 90 degrees (where 1 to 3 are preferable for thenumber of s from the viewpoint of calculation speed and calculationefficiency, and n signifies a degree in the above equation (r)).

Concretely, for example,

(in a case when the functional relation is a polynominal of an angle)

letting the polynominal be a quartic polynominal of the azimuth θ, and aat 0 degree, 45 degrees, 90 degrees be a(n, 0), a(n, 45), a(n, 90)respectively, a(n, θ) becomes the following equation (t):

a(n, θ)=a(n, 0)+(−11

·a(n, 0)+16·a(n, 45)−5

·a(n, 90))

·θ²/(4·90²)+(9·a(n, 0)−16

·a(n, 45)+7·a(n, 90))

·θ³/(4·90³)+(−2

·a(n, 0)+4

·a(n, 45)−2

·a(n, 90))

·θ⁴/(4·90⁴)  (t)

In this case, the above-mentioned control parameter in {circle around(4)} is one for the degree n of the distance r from the center and thecontrol parameter P1(n) is a(n, 45).

(in a case when the functional relation is not a polynominal of anangle, for example, is a trigonometric function)

a(n, θ) is expressed by the following equation (u), letting a be a(n, 0)and a(n, 90) when the azimuth θ is 0 degree and 90 degrees respectivelyin the following function and letting the control parameter which is onefor the degree n of the distance r from the center be P1(n) similarly tothe above:

a(n, θ)=a(n, 0)·cos² θ+a(n, 90)·sin² θ+P(1, n)·sin²(2·θ)   (u)

The equations (t), (u) both satisfy the above conditions {circle around(1)} to {circle around (4)}.

Thus, there exist various equations satisfying the above conditions{circle around (1)} to {circle around (4)}.

EXAMPLE 1

In Example 1, a spectacle lens is designed using the evaluation functionon visual acuity of the present invention, and the outline of thedesigning procedure thereof will be explained below.

(Step 1): To Set a Basic Design Lens Form of Front and Rear RefractiveSurfaces

In this example, a bi-aspherical lens form which has the highest degreeof freedom of designing is selected, with the front surface being anaspherical surface which is axially symmetrical and expressed by theabove equation (q) and with the rear surface being an aspherical surfaceexpressed by the above equation (r).

(Step 2): To Set Fixed Conditions and Variable Conditions of a ShapeDetermining Factor Parameter

The design conditions are, in the prescription values, a sphericaldiopter is −7.00 D, a cylindrical diopter is −2.00 D, a refractive index(ne) is 1.7, a lens diameter is 75 mm, and a lens center thickness is 1mm, as shown in FIG. 7.

In the above aspherical surface equations (q) and (r), k(θ) is 0 and theequation (t) is applied to a(n, θ).

Note that coefficients in the equations are as shown in FIG. 7.

(Step 3): To Set the Merit Function and a Target Value of theOptimization Calculation

The above equation (p) is used for the merit function and its conditionis a=1, b=0, c=0, and u=1.

The equation (j) is used for the equation of the visual acuityevaluation function logMAR and its condition is X=2.

(Step 4): Optimization Calculation

Based on set lens evaluation points, their evaluation is made using theaforesaid merit function by the ray tracing method, optical performanceis evaluated, simulation calculation is repeated by operatingtransformation parameters constituting the lens refractive surface untilthe predetermined target value is obtained, and the optimizationcalculation is carried out.

At this time, an optimal solution is calculated under the condition thatthe curvature of the front surface does not become negative(incidentally, a lens whose curvature of the front surface becomesnegative is described in Prior art 1, but it cannot be said to beaesthetically optimal since a reflected light is strong).

In this example, the final refractive surface shape is determined byfixing the design condition that the front surface is aspherical and byvarying the shape of the rear surface so as to satisfy the prescriptionvalues. Obtained lens data (final lens performance data after theoptimization is finished) are shown in Table 1 in FIG. 3.

Further, the distribution of the logMAR visual acuity values in Example1 in the case of the lens data in FIG. 3 is shown in FIG. 5.

64% of a thin portion in the lens center part produces preferable visualacuity whose logMAR visual acuity value is 0.2 or lower.

A Percival Form lens in which the curvature of field is reduced underthe same condition as that of Example 1 is shown for comparison.

Obtained lens data and the distribution of the logMAR visual acuityvalues are shown in Table 2 in FIG. 4 and FIG. 6 respectively.

The curvature of field of this lens is preferable, but 56% of the thinportion of the lens center part produces the preferable visual acuitywhose logMAR visual acuity value is 0.2 or lower.

Thus, it is clear that the preferable visual acuity range is obtained inFIG. 5, compared with that in FIG. 6, and the control of the evaluationfunction of visual acuity can sufficiently be performed so that theexpected effect is obtained.

EXAMPLE 2

In Example 2, an evaluation function on the residual distortionaberration DIST is further added to Example 1 to design a spectaclelens. Since the lens does not produce visual acuity and the optimalsolution cannot be obtained when only the residual distortion aberrationDIST is used in the aforesaid merit function equation (p), the logMARvisual acuity value and the residual distortion aberration DIST arebalanced in the equation (p).

In the equation (p), a=1, b=0, c=0.02, u=1, and w=1, and the equation(j) is used for the equation for the visual acuity evaluation functionlogMAR.

The equations (q), (r) are used for the bi-aspherical surface equation,k(θ) is 0, and the equation (t) is applied for a(n, θ).

The data in FIG. 5 in Example 1 are used for the front surface. Thoughthis is not a suitable condition for greatly improving the residualdistortion aberration DIST since a fixed condition is set for the frontand rear surfaces, the optimization calculation is done under the abovecondition since it is indicated that the residual distortion aberrationDIST can be controlled within a certain range. Obtained lens data areshown in Table 3 in FIG. 7.

FIG. 8 is a table showing the distribution of the residual distortionaberration DIST in the first quadrant. The lowest right end is the lenscenter, where the residual distortion aberration DIST is 0. Thehorizontal axis is a lens exit angle in the lens S axis direction, whichis shown for every 3 degree pitch, and similarly, the vertical axis isthe same in the lens C axis direction.

FIG. 9 shows, as a comparison example, the distribution of the residualdistortion aberration DIST under the condition in FIG. 5 in Example 1 inwhich the residual distortion aberration DIST is not evaluated as anevaluation function. The final values of the horizontal axis and thevertical axis in FIG. 8 are 43% and 60%, and the final values of thehorizontal axis and the vertical axis in FIG. 9 are 44% and 63%. Since asmaller value signifies more preferable state in this case, it is clearthat the control of the evaluation function for the residual distortionaberration DIST can be sufficiently performed and the expected effect isobtained.

Incidentally, a distribution view of the logMAR visual acuity under thecondition in FIG. 7 is shown in FIG. 10. The range where the logMARvisual acuity value is 0.2 or lower is 53%, and in improving theresidual distortion aberration DIST and the logMAR visual acuity value,they are in a trade-off correlation in which, when one value isimproved, the other value is lowered.

However, since sway is usually sensed at a peripheral portion, it isalso possible to improve the residual distortion aberration DIST in sucha manner that the distribution of the weights (u, v, w) at therespective evaluation points in the aforesaid merit function equation(q) is devised so as to give a higher weight to the logMAR visual acuityvalue in the center portion and to sacrifice the logMAR visual acuityvalue in the peripheral portion.

The merit function including the visual acuity evaluation functionaccording to the present invention is used for the bi-aspherical typelens having a single focus in this example. However, since the technicalstructure of the invention is characterized in that the visual acuityevaluation function is used as the evaluation function of the meritfunction used in the optimization calculation, it is not limited by therefractive shape of the lens surface, and can be used in designing ofall lenses including progressive refracting surfaces.

For example, in a progressive-power lens, other factors such as adistance portion, a near portion, a progressive zone are added besidesweighting on the lens central portion and peripheral portion, which isused in a case of ordinary lenses, and near vision weighted design, farvision weighted design, intermediate vision weighted design, and so onare also added to the object of the designing. However, since theprogressive-power lens uses the aspherical lens surface similarly tothis example when classified in terms of a lens surface, the presentinvention is applicable to the progressive-power lens by making themerit function according to the present invention correspond to theobject of its designing, appropriately setting the weight distributionat the evaluation points, setting target diopter and a target distortionaberration, and changing these design items.

The present invention is especially useful for the designing in whichthe Listing's Law is taken into consideration since an accuratesimulation can be carried out.

Furthermore, the same thing can be said for a lens whose rear surface isa fusion surface of an aspherical surface and an astigmatic surface.

In this example, data in Optmetric Monthly, November: 31-32 1981:written by Robert N. Kleinstein are used as a paper on the processing inthe retina and the brain regarding visual acuity. The present invention,however, is not limited to this, and any data can be used and the visualacuity evaluation function included in the present invention can bederived from the data, as long as they are data on the visual acuitymeasured value in which, for example, visual acuity and diopter arecorrelated.

Furthermore, in a manufacturing method, in the case of, for example, thebi-aspherical lens in this example, the front surface is made to be anaxially symmetrical aspherical surface and the rear surface is made tobe an aspherical lens of the free curved surface, so that asemi-finished lens can be used, which is effective in terms of time andcost. In other words, when a plurality of axially symmetrical asphericallens having a predetermined common base curve are prepared in advance asdescribed above, the semi-finished lens is first selected according tothe prescription after receipt of order, and thereafter, its rearsurface is designed, it is more advantageous than to design a convexsurface and a concave surface after each receipt of order and prepare afinished lens.

Moreover, by the aforesaid fixing of the design, it becomes possible toprepare finished products in advance in stock according to theprescription.

Industrial Availability

As detailed above, in contrast to a spectacle lens designing in a priorart in which the performance of a spectacle lens is evaluated only withindexes such as an optical amount on the retina and aberrations based onthe technical idea that the performance of the spectacle lens isimproved as optical performance is made higher, it becomes possible todesign a spectacle lens based on a simulation on a living human body, inwhich the viewpoints of the processing in the retina and the brain andof an eyeball motion are taken into consideration, and a spectacle lenswith higher performance can be obtained.

What is claimed is:
 1. A spectacle lens designing method in which motionof an eyeball (Listing's Law) is taken into consideration, the methodcomprising: performing an optimization calculation process for designinga spectacle lens using a merit function which includes a visual acuityevaluation function (logMAR) derived from a visual acuity measured valueV, where the visual acuity evaluation function (logMAR) is expressed bythe following equation (1) the visual acuity evaluation function(logMAR)=log₁₀(1/V (curvature of field, residual astigmatism))  (1), wherein a curvature of field is an aberration of the spectacle lens anda residual astigmatism is an astigmatism defined from spectacle lensdesigning in which Listing's Law is considered.
 2. A spectacle lensdesigning method according to claim 1, wherein said visual acuitymeasured value V is V=2^(−x·K) (where K={(residual S diopter+residual Cdiopter/2)²+(residual C diopter/2)²}^(1/2) and X is a coefficientbetween 0.5 and 2 according to actual measurement data), and said visualacuity evaluation function (logMAR) is expressed by the followingequation (2) which is an approximate equation: the visual acuityevaluation function (logMAR)=X×log₁₀2×{curvature of field²+(residualastigmatism/2)²}^(1/2)  (2).
 3. A spectacle lens designing methodaccording to claim 1, wherein said merit function includes an evaluationfunction on a distortion aberration (residual distortion aberrationDIST) and said evaluation function is expressed by the followingequation (3): residual distortion aberration DIST=Sign×100×(absolutevalue of residual visual angle magnification/absolute value of centralvisual angle magnification M ₀)  (3) where: the residual visual anglemagnification is the distortion aberration defined from the spectaclelens designing in which Listing's Law is considered; and Sign is apositive/negative sign.
 4. A spectacle lens designing method accordingto claim 1, wherein said merit function is used to perform anoptimization calculation process for designing a bi-aspherical lens inwhich a front surface is an axially symmetrical aspherical surface and arear surface is an aspherical surface expressed by the followingequation (4): $\begin{matrix}{{Z2} = {{c(\theta)} \cdot {r^{2}/( {1 + \sqrt{ {1 - {( {1 + {k(\theta)}} ) \cdot {c(\theta)}^{2} \cdot r^{2}}} )} + {\sum\limits_{n}{{a( {n,\theta} )} \cdot r^{n}}}} }}} & (4)\end{matrix}$

where: c(ζ), k(θ) are functions for an azimuth θ; a(n, θ) is a functionfor an n degree of a distance r and the azimuth θ; as for a definitiondomain of the azimuth θ, 0 degrees to 90 degrees represents 0 degrees to360 degrees due to plane symmetry of an astigmatic lens; c(θ) is acurvature of a lens center and is expressed by the following equation(5) based on Euler's theorem, letting a curvature of a spectacleprincipal meridian in a Gaussian curve theorem be c(0) at 0 degrees andc(90) at 90 degrees, wherein, 0 degrees is a spherical diopter axis and90 degrees is a cylinder diopter axis; c(θ)=c(0)° cos² θ+c(90)°sin²θ  (5); c(θ) is a curvature of a lens center and is expressed by thefollowing equation (5) based on Euler's theorem, letting a curvature ofa spectacle principal meridian in a Gaussian curve theorem be c(0) at 0degrees and c(90) at 90 degrees, wherein, 0 degrees is a sphericaldiopter axis and 90 degrees is a cylinder is a cylinder diopter axis;c(θ)=c(0)° cos² θ+c(90)° sin²θ  (5) k(θ), which is similar to said c(θ),represents an equation in which the sign c is replaced by the sign k insaid equation (5); and a(n, θ) satisfies requirements of planecontinuity and plane symmetry, is a surface further satisfying arequirement of a surface which is capable of controlling an aberrationdue to an angle deviation which occurs due to the Listing's Law, andfurther satisfies the following conditions 1 to 4: 1: having afunctional relation of the azimuth θ from 0 degree to 90 degrees; 2: alinear differential coefficient of the azimuth θ is 0 from 0 degree to90 degrees; 3: a high degree differential coefficient is continuous; and4: having a control parameter group Ps(n) which is capable ofcontrolling a value of a(n, θ) at an angle θ of a function between theazimuths of 0 degree and 90 degrees (where 1 to 3 are preferable for thenumber of s, and n signifies a degree in said equation (4)).
 5. Aspectacle lens designing method according to claim 4, wherein a(n, θ) insaid equation (4) is expressed by the following equation (6) which is aquartic polynominal of the azimuth θ, letting a be a(n, 0), a(n, 45),and a(n, 90) when azimuth θ is 0 degree, 45 degrees, and 90 degreesrespectively: a(n, θ)=a(n, 0) +(−11°a(n, 0) +16°a(n, 45) −5°a(n,90))°θ²/(4°90²) +(9°a(n, 0) −16°a(n, 45) +7°a(n, 90))°θ³/(4°90³)+(−2°a(n, 0) +4°a(n, 45) −2°a(n, 90))° θ⁴/(4°90⁴)  (6) where a controlparameter is one for the degree n of the distance r from the center anda control parameter P1(n) is a(n, 45).
 6. A spectacle lens designingmethod according to claim 4, wherein a(n, θ) in said equation (4) isexpressed by the following equation (7), where a is a(n, 0) and a(n, 90)when the azimuth θ is 0 degrees and 90 degrees respectively: a(n,θ)=a(n, 0)° cos² θ+a(n, 90)° sin² θ+P 1(n)° sin²(2°θ)  (7), where thecontrol parameter is one control parameter P1(n) for the degree n of thedistance r from the center.
 7. A spectacle lens, being designed by thespectacle lens designing method according to claim
 1. 8. A spectaclelens designing method according to claim 2, wherein said merit functionis used to perform an optimization calculation process for designing abi-aspherical lens in which a front surface is an axially symmetricalaspherical surface and a rear surface is an aspherical surface expressedby the following equation (4): $\begin{matrix}{{Z2} = {{c(\theta)} \cdot {r^{2}/( {1 + \sqrt{ {1 - {( {1 + {k(\theta)}} ) \cdot {c(\theta)}^{2} \cdot r^{2}}} )} + {\sum\limits_{n}{{a( {n,\theta} )} \cdot r^{n}}}} }}} & (4)\end{matrix}$

where: c(θ), k(θ) are functions for an azimuth θ; a(n, θ) is a functionfor an n degree of a distance r and the azimuth θ; as for a definitiondomain of the azimuth θ, 0 degrees to 90 degrees represents 0 degrees to360 degrees due to plane symmetry of an astigmatic lens; c(θ) is acurvature of a lens center and is expressed by the following equation(5) based on Euler's theorem, letting a curvature of a spectacleprincipal meridian in a Gaussian curve theorem be c(0) at 0 degrees andc(90) at 90 degrees, wherein, 0 degrees is a spherical diopter axis and90 degrees is a cylinder diopter axis; c(θ)=c(0)° cos² θ+c(90)°sin²θ  (5); k(θ), which is similar to said c(θ), represents an equationin which the sign c is replaced by the sign k in said equation (5); anda(n, θ) satisfies requirements of plane continuity and plane symmetry,is a surface further satisfying a requirement of a surface which iscapable of controlling an aberration due to an angle deviation whichoccurs due to the Listing's Law, and further satisfies the followingconditions 1 to 4: 1: having a functional relation of the azimuth θ from0 degrees to 90 degrees; 2: a linear differential coefficient of theazimuth θ is 0 from 0 degrees to 90 degrees; 3: a high degreedifferential coefficient is continuous; and 4: having a controlparameter group Ps(n) which is capable of controlling a value of a(n, θ)at an angle θ of a function between the azimuths of 0 degrees and 90degrees (where 1 to 3 are preferable for the number of s, and nsignifies a degree in said equation (4)).
 9. A spectacle lens designingmethod according to claim 3, wherein said merit function is used toprocess an optimization calculation for designing a bi-aspherical lensin which a front surface is an axially symmetrical aspherical surfaceand a rear surface is an aspherical surface expressed by the followingequation (4): $\begin{matrix}{{Z2} = {{c(\theta)} \cdot {r^{2}/( {1 + \sqrt{ {1 - {( {1 + {k(\theta)}} ) \cdot {c(\theta)}^{2} \cdot r^{2}}} )} + {\sum\limits_{n}{{a( {n,\theta} )} \cdot r^{n}}}} }}} & (4)\end{matrix}$

where: c(θ), k(θ) are functions for an azimuth θ; a(n, θ) is a functionfor an n degree of a distance r and the azimuth θ; as for a definitiondomain of the azimuth θ, 0 degrees to 90 degrees represents 0 degrees to360 degrees due to plane symmetry of an astigmatic lens; k(θ), which issimilar to said c(θ), represents an equation in which the sign c isreplaced by the sign k in said equation (5); and a(n, θ) satisfiesrequirements of plane continuity and plane symmetry, is a surfacefurther satisfying a requirement of a surface which is capable ofcontrolling an aberration due to an angle deviation which occurs due tothe Listing's Law, and further satisfies the following conditions 1 to4: 1: having a functional relation of the azimuth θ from 0 degrees to 90degrees; 2: a linear differential coefficient of the azimuth θ is 0 from0 degrees to 90 degrees; 3: a high degree differential coefficient iscontinuous; and 4: having a control parameter group Ps(n) which iscapable of controlling a value of a(n, θ) at an angle θ of a functionbetween the azimuths of 0 degrees and 90 degrees (where 1 to 3 arepreferable for the number of s, and n signifies a degree in saidequation (4)).
 10. A spectacle lens, being designed by the spectaclelens designing method according to claim
 2. 11. A spectacle lens, beingdesigned by the spectacle lens designing method according to claim 3.12. A spectacle lens, being designed by the spectacle lens designingmethod according to claim
 4. 13. A spectacle lens, being designed by thespectacle lens designing method according to claim
 5. 14. A spectaclelens, being designed by the spectacle lens designing method according toclaim 6.